Question: Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{a^2 - 2a - 35}{a^2 - 6a - 7}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{a^2 - 2a - 35}{a^2 - 6a - 7} = \dfrac{(a + 5)(a - 7)}{(a + 1)(a - 7)} $ Notice that the term $(a - 7)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a - 7)$ gives: $k = \dfrac{a + 5}{a + 1}$ Since we divided by $(a - 7)$, $a \neq 7$. $k = \dfrac{a + 5}{a + 1}; \space a \neq 7$